Time Series Analysis of the Hybrid Quantum-Classical Method of Simulation of High-Temperature Spin Dynamics

Abstract

Series expansion of the two-spin correlation functions is obtained in the context of the hybrid quantum-classical method of simulation of high-temperature spin dynamics. The resulting series are compared with the corresponding series for the exact correlation functions obtained from purely quantum dynamics.

Appendix: Infinite-Temperature Correlators of Arbitrary Order

In the calculation of the series expansion of the correlation functions in the hybrid case, one encounters the correlators of the form:

$$\begin{aligned} G^{(n)}_{{\mathscr {A}}_1,{\mathscr {A}}_2,\dotsc ,{\mathscr {A}}_n} = \left[ \langle \psi |{\mathscr {A}}_1|\psi \rangle \cdot \langle \psi |{\mathscr {A}}_2|\psi \rangle \cdot \cdots \cdot \langle \psi |{\mathscr {A}}_n|\psi \rangle \right] _\psi , \end{aligned}$$

(49)

where $[\cdots ]_\psi$ denotes the average over the infinite-temperature ensemble of normalized uniformly distributed wave-functions $|\psi \rangle$. If we represent the wave-function

$$\begin{aligned} |\psi \rangle = \sum \limits _{k=1}^D a_k |k\rangle , \end{aligned}$$

(50)

in terms of some full orthonormal basis ${|k\rangle }$, we can reformulate Eq. (49) as

$$\begin{aligned}&G^{(n)}_{{\mathscr {A}}_1,{\mathscr {A}}_2,\dotsc ,{\mathscr {A}}_n} \nonumber \\&\quad = \sum \limits _{i_1,j_1}\sum \limits _{i_2,j_2}\cdots \sum \limits _{i_n,j_n} \left[ a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\right] _\psi ({\mathscr {A}}_1)_{i_1,j_1} ({\mathscr {A}}_2)_{i_2,j_2}\cdots ({\mathscr {A}}_n)_{i_n,j_n}, \end{aligned}$$

(51)

where

$$\begin{aligned} ({\mathscr {A}}_p)_{i_p,j_p} = \langle i_p|{\mathscr {A}}_p| j_p\rangle . \end{aligned}$$

(A4)

Given a particular choice of orthonormal basis in representation (50), the correlators for arbitrary sets of observables are completely determined by the matrix elements of observables and by the tensors:

$$\begin{aligned} F^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n} \equiv \left[ a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\right] _\psi . \end{aligned}$$

(53)

Since the distribution of wave functions describing infinite temperature is invariant with respect to unitary transformations, the choice of the orthonormal basis $|k\rangle$ in representation (50) is arbitrary. As a consequence, the form of the tensors $F^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n}$ is identical irrespective of this choice.

In order to deduce the form of the tensors $F^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n}$ of arbitrary order, it is convenient to recast the Hilbert space average $[\cdots ]_{\psi }$ as a multidimensional integral over the expansion coefficients $a_i$ in representation (50):

$$\begin{aligned}&F^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n} \equiv \left[ a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\right] _\psi \nonumber \\&\quad = \frac{\int \prod \nolimits _{k=1}^D \mathrm{d}a^*_k \mathrm{d}a_k \ \delta \left( \sqrt{\sum \nolimits _{k=1}^D a^*_k a_k} -1\right) a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}}{\int \prod \nolimits _{k=1}^D \mathrm{d}a^*_k \mathrm{d}a_k\ \delta \left( \sqrt{\sum \nolimits _{k=1}^D a^*_k c_k} - 1\right) }, \end{aligned}$$

(54)

where $\delta (\cdots )$ is the Dirac delta-function. Here the measure of integration $\mathrm{d}\mu$ is defined in terms of the decomplexification of the Hilbert space:

$$\begin{aligned} d\mu = \prod _{i=1}^D \mathrm{d}a_i^*\mathrm{d}a_i = \prod _{p=1}^D \mathrm{d}{{\,\mathrm{Re}\,}}{a^*_i}\mathrm{d}{{\,\mathrm{Im}\,}}{a_i}. \end{aligned}$$

(55)

The denominator of Eq. (54) is the area $A_{2D-1}$ of the $(2D-1)$-dimensional unit hypersphere.

In order to evaluate the right-hand side of Eq. (54), let us introduce an auxiliary multidimensional integral:

$$\begin{aligned} I^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n} = \int \prod \limits _{k=1}^D \frac{\mathrm{d}a_k^*\mathrm{d}a_k}{\pi }\ e^{-\sum \limits _{k=1}^D a^*_ka_k} a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}. \end{aligned}$$

(56)

On the one hand, it is a simple Gaussian integral and can be evaluated in the closed form with the use of the Wick’s theorem (see, for example, [21, Chapter 1]) as

$$\begin{aligned} I^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n} = \sum \limits _{P} \delta _{i_1, Pj_1} \cdot \delta _{i_2, Pj_2}\cdot \cdots \cdot \delta _{i_n,Pj_n}, \end{aligned}$$

(57)

where the summation is performed over all the permutations P of n indices $\{j_1,j_2,\dotsc ,j_n\}$. On the other hand, we can split the integration into the integration over the surface of the $(2D-1)$-dimensional hyper-sphere of radius r with the subsequent integration over the radius of that hypersphere:

$$\begin{aligned}&\int \prod \limits _{k=1}^D \frac{\mathrm{d}a_k^*\mathrm{d}a_k}{\pi }\ e^{-\sum \limits _{k=1}^D a^*_ka_k} a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\nonumber \\&\quad =\int \limits _0^{+\infty } \mathrm{d}r\ r^{2D-1} \int \prod \limits _{k=1}^D \frac{\mathrm{d}a_k^*\mathrm{d}a_k}{\pi }\ \delta \left( \sqrt{\sum \limits _{k=1}^D a^*_k a_k} - r\right) e^{-\sum \limits _{k=1}^D a^*_ka_k} a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\nonumber \\&\quad =\left[ a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\right] _\psi \cdot \frac{A_{2D-1}}{\pi ^D}\int \limits _0^{+\infty } dr e^{-r^2} r^{2(D+n)-1}. \end{aligned}$$

(58)

Here we have used the fact that

$$\begin{aligned}&\int \prod \limits _{k=1}^D \frac{\mathrm{d}c_k^*\mathrm{d}c_k}{\pi }\ \delta \left( \sqrt{\sum \limits _{k=1}^D a^*_k c_k} - r\right) e^{-\sum \limits _{k=1}^D a^*_kc_k} a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\nonumber \\&\quad =e^{-r^2} \int \prod \limits _{k=1}^D \frac{\mathrm{d}c_k^*\mathrm{d}c_k}{\pi }\ \delta \left( \sqrt{\sum \limits _{k=1}^D a^*_k c_k} - r\right) a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\nonumber \\&\quad = e^{-r^2} r^n \int \prod \limits _{k=1}^D \frac{dc_k^*\mathrm{d}c_k}{\pi }\ \delta \left( \sqrt{\sum \limits _{k=1}^D a^*_k c_k} -1\right) a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n} \nonumber \\&\quad =e^{-r^2} r^n A_{2D-1}\left[ a^*_{i_1}a_{j_1}a^*_{i_2}a_{j_2}\cdots a^*_{i_n}a_{j_n}\right] _\psi . \end{aligned}$$

(59)

In Eq. (58),

$$\begin{aligned}&\frac{A_{2D-1}}{\pi ^D}\int \limits _0^{+\infty } \mathrm{d}r e^{-r^2} r^{2(D+n)-1} \nonumber \\&\quad = \frac{A_{2D-1}}{\pi ^D}\int \limits _0^{+\infty } \mathrm{d}(r^2/2) e^{-r^2} r^{2(D+n-1)} = \frac{\varGamma (D+n)A_{2D-1}}{2\pi ^D}, \end{aligned}$$

(60)

where $\varGamma (x)$ is the Euler’s gamma function, and

$$\begin{aligned} A_n \equiv \frac{2\pi ^{\frac{n+1}{2}}}{\varGamma (\frac{n+1}{2})} \end{aligned}$$

(61)

for odd n. Thus,

$$\begin{aligned} \frac{\varGamma (D+n)A_{2D-1}}{2\pi ^D} = \frac{\varGamma (D+n)}{\varGamma (D)}. \end{aligned}$$

(62)

Substituting Eq. (62) into Eq. (58) and comparing the result with Eq. (57), we, finally, get

$$\begin{aligned} F^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n} = \frac{\varGamma (D)}{\varGamma (D+n)} I^{i_1,i_2,\dotsc ,i_n}_{j_1,j_2,\dotsc ,j_n} = \frac{\varGamma (D)}{\varGamma (D+n)} \sum \limits _{P} \delta _{i_1, Pj_1} \cdot \delta _{i_2, Pj_2}\cdot \cdots \cdot \delta _{i_n,Pj_n}. \end{aligned}$$

(63)